The meantone family

This gives part of the family tree of the meantone family; I don't go
into cousins such as the various 11-limit versions of flattone or
dominant sevenths. In each limit, I propose giving the name "meantone"
to the temperament with the same TOP generators as 5-limit meantone.
The numbers before the colon give the nexial val with the base
meantone, plus or minus depending on whether adding or subtracting is
required. Also, the TM basis and the TOP octave and fourth.

meantone 81/80 comma
[1201.70, 504.13]

7-limit meantone family

0: <<1 4 10 4 13 12|| meantone {81/80, 126/125}
[1201.70, 504.13]

-5: <<1 4 5 4 5 0|| {15/14, 81/80}
[1185.31, 500.33]

-7: <<1 4 3 4 2 -4|| {21/20, 28/27}
[1214.25, 509.40]

-12: <<1 4 -2 4 -6 -16|| dominant seventh {36/35, 64/63}
[1195.23, 495.88]

-19: <<1 4 -9 4 -17 -32|| flattone {81/80, 525/512}
[1202.54, 507.14]

-31: <<1 4 -21 4 -36 -60|| {81/80, 65625/65536}
[1201.56, 503.96]

11-limit meantone family (no cousins)

0: <<1 4 10 -13 4 13 -24 12 -44 -71|| meantone {81/80, 126/125, 385/384}
[1201.70, 503.13]

19: <<1 4 10 6 4 13 6 12 0 -18|| {45/44, 56/55, 81/80}
[1198.56, 503.60]

31: <<1 4 10 18 4 13 25 12 28 16|| huygens {81/80, 99/98, 126/125}
[1201.61, 504.02]

Gene Ward Smith wrote:
> This gives part of the family tree of the meantone family; I don't go
> into cousins such as the various 11-limit versions of flattone or
> dominant sevenths. In each limit, I propose giving the name "meantone"
> to the temperament with the same TOP generators as 5-limit meantone.
> The numbers before the colon give the nexial val with the base
> meantone, plus or minus depending on whether adding or subtracting is
> required. Also, the TM basis and the TOP octave and fourth.
> > meantone 81/80 comma > [1201.70, 504.13]
> > 7-limit meantone family
> > 0: <<1 4 10 4 13 12|| meantone {81/80, 126/125}
> [1201.70, 504.13]

Hmm, I must have the brackets backward on my web page; I have meantone as [1, 4, 10, 4, 13, 12>. I can never remember which way they go. But is there any significance to the double brackets?

(For anyone who might not be on the tuning list, the page I'm referring to is http://www.io.com/~hmiller/music/zireen-music.html)

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:

> Hmm, I must have the brackets backward on my web page; I have
meantone
> as [1, 4, 10, 4, 13, 12>. I can never remember which way they go.
But is
> there any significance to the double brackets?

The double brackets signal that it is a bival; wedgies for linear
temperaments are always wedge products of two vals.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Gene Ward Smith wrote:
> > This gives part of the family tree of the meantone family; I
don't go
> > into cousins such as the various 11-limit versions of flattone or
> > dominant sevenths. In each limit, I propose giving the
name "meantone"
> > to the temperament with the same TOP generators as 5-limit
meantone.
> > The numbers before the colon give the nexial val with the base
> > meantone, plus or minus depending on whether adding or
subtracting is
> > required. Also, the TM basis and the TOP octave and fourth.
> >
> > meantone 81/80 comma
> > [1201.70, 504.13]
> >
> > 7-limit meantone family
> >
> > 0: <<1 4 10 4 13 12|| meantone {81/80, 126/125}
> > [1201.70, 504.13]
>
> Hmm, I must have the brackets backward on my web page; I have
meantone
> as [1, 4, 10, 4, 13, 12>.

These can't both be right, because taking the complement changes some
of the signs. And in fact, your web page is incorrect. The bival
bracket points to the left but has the numbers you cite. The bimonzo
bracket points to the right but the sign on the second and fifth
elements would have to be reversed. You should compute these yourself
to see it. Try computing the wedge product of the monzos for 81/80
and 126/125, for example.

> I can never remember which way they go. But is
> there any significance to the double brackets?

To remind you that these are bivectors rather than vectors.

Paul Erlich wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> >>Hmm, I must have the brackets backward on my web page; I have > > meantone > >>as [1, 4, 10, 4, 13, 12>.
> > > These can't both be right, because taking the complement changes some > of the signs. And in fact, your web page is incorrect. The bival > bracket points to the left but has the numbers you cite. The bimonzo > bracket points to the right but the sign on the second and fifth > elements would have to be reversed. You should compute these yourself > to see it. Try computing the wedge product of the monzos for 81/80 > and 126/125, for example.

I've since updated the web page (I was intending the bival bracket but had the brackets pointing the wrong way). Although it seems that I never got around to uploading the corrected version with double brackets. Well, it's up now.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Paul Erlich wrote:
> > --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
> >>Hmm, I must have the brackets backward on my web page; I have
> >
> > meantone
> >
> >>as [1, 4, 10, 4, 13, 12>.
> >
> >
> > These can't both be right, because taking the complement changes
some
> > of the signs. And in fact, your web page is incorrect. The bival
> > bracket points to the left but has the numbers you cite. The
bimonzo
> > bracket points to the right but the sign on the second and fifth
> > elements would have to be reversed. You should compute these
yourself
> > to see it. Try computing the wedge product of the monzos for
81/80
> > and 126/125, for example.
>
> I've since updated the web page (I was intending the bival bracket
but
> had the brackets pointing the wrong way). Although it seems that I
never
> got around to uploading the corrected version with double brackets.
> Well, it's up now.

Your webpage still seems wrong. You say:

"Take for instance the commas 81/80 and 126/125: you can wedge them
to get <<1, 4, 10, 4, 13, 12]]."

But this is not correct. Putting aside the direction of the brackets,
the numbers are simply not right. Either two or four of the signs
must be reversed, depending on the order in which 81/80 and 126/125
are wedged together. Try it!

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Your webpage still seems wrong. You say:
>
> "Take for instance the commas 81/80 and 126/125: you can wedge them
> to get <<1, 4, 10, 4, 13, 12]]."
>
> But this is not correct.

It's correct shorthand; given that we see a wedgie (normalized bival)
above, it means we wedge the monzos for 81/80 and 126/125, take the
complement, and normalize to a wedgie. In other words, you can wedge
the two intervals and get the wedgie.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Your webpage still seems wrong. You say:
> >
> > "Take for instance the commas 81/80 and 126/125: you can wedge
them
> > to get <<1, 4, 10, 4, 13, 12]]."
> >
> > But this is not correct.
>
> It's correct shorthand; given that we see a wedgie (normalized
bival)
> above, it means we wedge the monzos for 81/80 and 126/125, take the
> complement, and normalize to a wedgie. In other words, you can wedge
> the two intervals and get the wedgie.

Look at the sentence Herman had on his webpage immediately before
this one, which includes the term "wedge product". In the context,
the statement is simply not correct. Herman would have to add "and
take the complement", at a minimum.

Paul Erlich wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> > wrote:
> >>--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> > > wrote:
> >>>Your webpage still seems wrong. You say:
>>>
>>>"Take for instance the commas 81/80 and 126/125: you can wedge > > them > >>>to get <<1, 4, 10, 4, 13, 12]]."
>>>
>>>But this is not correct.
>>
>>It's correct shorthand; given that we see a wedgie (normalized > > bival)
> >>above, it means we wedge the monzos for 81/80 and 126/125, take the
>>complement, and normalize to a wedgie. In other words, you can wedge
>>the two intervals and get the wedgie.
> > > Look at the sentence Herman had on his webpage immediately before > this one, which includes the term "wedge product". In the context, > the statement is simply not correct. Herman would have to add "and > take the complement", at a minimum.

Well, it's a tricky situation. I need to explain just enough about wedgies so that visitors to the page have at least a rough idea of what those strange numbers by the temperament names are and why they're of any use, without introducing all sorts of other concepts that aren't relevant to the page. After all, I'm using them more or less as unique "index numbers" for temperaments, for cross-reference purposes. But I can see how the description might be misleading. I do remark that wedging commas isn't the same as wedging maps (i.e., monzos vs. vals); a more in-depth explanation would reveal that the difference involves taking the complement, but I didn't want to get into that much detail.

How about changing "you can wedge them to get <<1, 4, 10, 4, 13, 12]]" to something like "you can wedge them and ultimately end up with <<1, 4, 10, 4, 13, 12]]"? Or would it be better to add more detail and go through a process like "You can represent 81/80 as [-4 4 -1 0> and 126/125 as [1 2 -3 1>; taking the wedge product gives a result of [[12, -13, 4, 10, -4, 1>> which after some further manipulation gives a normalized result of <<1, 4, 10, 4, 13, 12]]"?

I guess really the only way around it is to write a page on linear temperament theory and link to that. But that's another project for a later date.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> This gives part of the family tree of the meantone family; I don't
go
> into cousins such as the various 11-limit versions of flattone or
> dominant sevenths. In each limit, I propose giving the
name "meantone"
> to the temperament with the same TOP generators as 5-limit meantone.
> The numbers before the colon give the nexial val with the base
> meantone, plus or minus depending on whether adding or subtracting
> is required.

Here's where I am stuck. I can see these values are all meantone ets,
but what do you mean exactly be "nexial val with the base meantone?"

Thanks.

> Also, the TM basis and the TOP octave and fourth.

>
> meantone 81/80 comma
> [1201.70, 504.13]
>
> 7-limit meantone family
>
> 0: <<1 4 10 4 13 12|| meantone {81/80, 126/125}
> [1201.70, 504.13]
>
> -5: <<1 4 5 4 5 0|| {15/14, 81/80}
> [1185.31, 500.33]
>
> -7: <<1 4 3 4 2 -4|| {21/20, 28/27}
> [1214.25, 509.40]
>
> -12: <<1 4 -2 4 -6 -16|| dominant seventh {36/35, 64/63}
> [1195.23, 495.88]
>
> -19: <<1 4 -9 4 -17 -32|| flattone {81/80, 525/512}
> [1202.54, 507.14]
>
> -31: <<1 4 -21 4 -36 -60|| {81/80, 65625/65536}
> [1201.56, 503.96]
>
>
> 11-limit meantone family (no cousins)
>
> 0: <<1 4 10 -13 4 13 -24 12 -44 -71|| meantone {81/80, 126/125,
385/384}
> [1201.70, 503.13]
>
> 19: <<1 4 10 6 4 13 6 12 0 -18|| {45/44, 56/55, 81/80}
> [1198.56, 503.60]
>
> 31: <<1 4 10 18 4 13 25 12 28 16|| huygens {81/80, 99/98, 126/125}
> [1201.61, 504.02]

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > This gives part of the family tree of the meantone family; I
don't
> go
> > into cousins such as the various 11-limit versions of flattone or
> > dominant sevenths. In each limit, I propose giving the
> name "meantone"
> > to the temperament with the same TOP generators as 5-limit
meantone.
> > The numbers before the colon give the nexial val with the base
> > meantone, plus or minus depending on whether adding or
subtracting
> > is required.
>
> Here's where I am stuck. I can see these values are all meantone
ets,
> but what do you mean exactly be "nexial val with the base meantone?"
>
> Thanks.

Is it the difference between c3 values (against the base 0: meantone)
in the 7-limit and c4 values in the 11-limit (with the 0: meantone)?

And this signifies p2,p7 and p2,p11 wedge values respectively...
>
> > Also, the TM basis and the TOP octave and fourth.
>
>
>
> >
> > meantone 81/80 comma
> > [1201.70, 504.13]
> >
> > 7-limit meantone family
> >
> > 0: <<1 4 10 4 13 12|| meantone {81/80, 126/125}
> > [1201.70, 504.13]
> >
> > -5: <<1 4 5 4 5 0|| {15/14, 81/80}
> > [1185.31, 500.33]
> >
> > -7: <<1 4 3 4 2 -4|| {21/20, 28/27}
> > [1214.25, 509.40]
> >
> > -12: <<1 4 -2 4 -6 -16|| dominant seventh {36/35, 64/63}
> > [1195.23, 495.88]
> >
> > -19: <<1 4 -9 4 -17 -32|| flattone {81/80, 525/512}
> > [1202.54, 507.14]
> >
> > -31: <<1 4 -21 4 -36 -60|| {81/80, 65625/65536}
> > [1201.56, 503.96]
> >
> >
> > 11-limit meantone family (no cousins)
> >
> > 0: <<1 4 10 -13 4 13 -24 12 -44 -71|| meantone {81/80, 126/125,
> 385/384}
> > [1201.70, 503.13]
> >
> > 19: <<1 4 10 6 4 13 6 12 0 -18|| {45/44, 56/55, 81/80}
> > [1198.56, 503.60]
> >
> > 31: <<1 4 10 18 4 13 25 12 28 16|| huygens {81/80, 99/98, 126/125}
> > [1201.61, 504.02]